文章基本信息

标题：Pinning down the Strong Wilber 1 Bound for Binary Search Trees
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作者：Parinya Chalermsook ; Julia Chuzhoy ; Thatchaphol Saranurak 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2020
卷号：176
页码：33:1-33:21
DOI：10.4230/LIPIcs.APPROX/RANDOM.2020.33
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：The dynamic optimality conjecture, postulating the existence of an O(1)-competitive online algorithm for binary search trees (BSTs), is among the most fundamental open problems in dynamic data structures. Despite extensive work and some notable progress, including, for example, the Tango Trees (Demaine et al., FOCS 2004), that give the best currently known O(log log n)-competitive algorithm, the conjecture remains widely open. One of the main hurdles towards settling the conjecture is that we currently do not have approximation algorithms achieving better than an O(log log n)-approximation, even in the offline setting. All known non-trivial algorithms for BSTâs so far rely on comparing the algorithmâs cost with the so-called Wilberâs first bound (WB-1). Therefore, establishing the worst-case relationship between this bound and the optimal solution cost appears crucial for further progress, and it is an interesting open question in its own right. Our contribution is two-fold. First, we show that the gap between the WB-1 bound and the optimal solution value can be as large as Î©(log log n/ log log log n); in fact, we show that the gap holds even for several stronger variants of the bound. Second, we provide a simple algorithm, that, given an integer D > 0, obtains an O(D)-approximation in time exp (O (n^{1/2^{Î©(D)}}log n)). In particular, this yields a constant-factor approximation algorithm with sub-exponential running time. Moreover, we obtain a simpler and cleaner efficient O(log log n)-approximation algorithm that can be used in an online setting. Finally, we suggest a new bound, that we call the Guillotine Bound, that is stronger than WB-1, while maintaining its algorithm-friendly nature, that we hope will lead to better algorithms. All our results use the geometric interpretation of the problem, leading to cleaner and simpler analysis.
关键词：Binary search trees; Dynamic optimality; Wilber bounds